Question

Let M be a point of contact of two internally touching circles. Let line AMB be their common tangent. The chord CD of the bigger circle touches the smaller circle at the point N. Chord CM and chord DM of the bigger circler intersect the smaller circle at the points P and R respectively.
Prove that `/_ CMN ~= /_ DMN`.
Construction `:` Draw seg NR.

Answer 1

CD si a tangent to the smaller circle at the point N and NR is a chord of the smaller circle.
`:. /_ DNR ~= /_ NMR ` …(Tangent secant theorem ) …(1)
AB is a tangent to the lager circle at the point M and CM is th echord of the larger circle.
`:. /_ CMA ~= /_ CDM ` ...(Tangent secant theorem )
i.e., `/_ CMA ~= /_D ` ...(2)
AB is a tangent to the smaller circle at the point M and NM is a chord of the smaller circle .
`:. /_ NMA ~= /_ NRM ` ....(Tangent secant theorem ) ...(3)
For `Delta NDR, /_ NRM ` is the exterior theorem,
`/_ NRM = /_ D + /_ DNR ` ...(4)
`:. /_ NMA = /_ CMA + /_ NMR` ........[From (3), (2) and (1) ] ....(5)
`/_ NMA = /_ NMC + /_ CMA ` ....(Angle addition postulate ) ...(6)
From (5) and (6)
`/_ NMC = /_ NMR `
`:. /_NMC = /_ NMD ` ...( `/_ NMR ` is the same as `/_ NMD `)
i.e., `/_ CMN ~= /_ DMN `.